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Theorem pnfnre 7814
Description: Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
pnfnre  |- +oo  e/  RR

Proof of Theorem pnfnre
StepHypRef Expression
1 cnex 7751 . . . . . 6  |-  CC  e.  _V
21uniex 4359 . . . . 5  |-  U. CC  e.  _V
3 pwuninel2 6179 . . . . 5  |-  ( U. CC  e.  _V  ->  -.  ~P U. CC  e.  CC )
42, 3ax-mp 5 . . . 4  |-  -.  ~P U. CC  e.  CC
5 df-pnf 7809 . . . . 5  |- +oo  =  ~P U. CC
65eleq1i 2205 . . . 4  |-  ( +oo  e.  CC  <->  ~P U. CC  e.  CC )
74, 6mtbir 660 . . 3  |-  -. +oo  e.  CC
8 recn 7760 . . 3  |-  ( +oo  e.  RR  -> +oo  e.  CC )
97, 8mto 651 . 2  |-  -. +oo  e.  RR
109nelir 2406 1  |- +oo  e/  RR
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1480    e/ wnel 2403   _Vcvv 2686   ~Pcpw 3510   U.cuni 3736   CCcc 7625   RRcr 7626   +oocpnf 7804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-un 4355  ax-cnex 7718  ax-resscn 7719
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-nel 2404  df-rex 2422  df-rab 2425  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-uni 3737  df-pnf 7809
This theorem is referenced by:  renepnf  7820  nn0nepnf  9055  xrltnr  9573  pnfnlt  9580  xnn0lenn0nn0  9655  inftonninf  10221
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